New Jersey Gov. Chris Christie enters the Borough Hall in Fort Lee to apologize to Mayor Mark Sokolich on Thursday.

The New Jersey Assemblyman leading the investigation into the George Washington Bridge controversy said Sunday that state lawmakers could consider impeaching Gov. Chris Christie if he was involved in the incident or any possible coverup.

Democratic Assemblyman John Wisniewski was careful, though, to say that no direct connection has been found between Mr. Christie and the lane closures, which the governor said Thursday may have been a “political vendetta” by his aides against the mayor of Fort Lee, N.J.

When first asked about the possibility of impeachment, Mr. Wisniewski said, “I think we are a little early on that.”

But if it turns out that Mr. Christie “was approving the actions taken of his senior staff, that raises serious questions that the Assembly ought to look at,” Mr. Wisniewski said on CBS’s “Face the Nation.” “The assembly has the ability to do articles of impeachment.”

A spokesman for Mr. Christie didn’t respond to a request for comment Sunday concerning impeachment.

Local access lanes to the bridge were closed for about five days—ending Sept. 13—causing severe traffic jams in Fort Lee, the borough that sits at the span’s western end. On Saturday, the Journal reported that a cabinet-level member of Mr. Christie’s administration was informed about the lane closures the day they were reversed.

Thousands of pages of documents subpoenaed by the state Assembly that were released last week showed that Mr. Christie’s deputy chief of staff, Bridget Anne Kelly, on Aug. 13 wrote “Time for some traffic problems in Fort Lee” to David Wildstein, a Christie ally at the Port Authority of New York and New Jersey. Mr. Wildstein replied, “Got it.” Mr. Christie fired Ms. Kelly last week and Mr. Wildstein resigned in December.

Democrats speculated that the closures were meant as punishment for Fort Lee mayor Mark Sokolich, a Democrat who didn’t endorse Mr. Christie for re-election in 2013. Christie allies at the Port Authority had previously said the closures were part of a traffic study—but those statements are now in doubt.

Most of the Sunday news talk shows touched on the controversy, with pundits and politicians discussing the political damage the controversy may have already caused Mr. Christie and how it impacts any presidential ambitions.

Republicans said that Mr. Christie should escape the damage from the scandal.

Former New York Mayor Rudy Giuliani, one of Mr. Christie’s most steadfast Republican supporters, said he believes the governor’s explanation that he didn’t know that his staff and allies on the Port Authority of New York and New Jersey were involved in the matter. Mr. Christie was running for re-election at the time of the closures.

“He was in campaign mode at the time,” Mr. Giuliani said on ABC’s “This Week.” “During campaign mode, you miss a lot of things. You’re not paying as much attention.”

The governor handled it well with a nearly two-hour news conference Thursday, Mr. Giuliani said. “I think it’s pretty darn credible. He wouldn’t make this blanket denial unless it is not true,” he said.

Still, Mr. Giuliani said that Mr. Christie put his entire political career at risk if any evidence surfaces that he knew about the lane closures.

“If, for some reason, it’s not true, the man has put his political career completely at risk if it turns out there is some evidence that he knew about it,” Giuliani added. “He’s taken the complete risk that his political career is over. I don’t think he’d do that if there’s any suggestion he knew about this.”

Republican National Committee Chairman Reince Priebus also said Mr. Christie’s news conference helped reduce political fallout. “Americans are a very forgiving people,” he said on NBC’s “Meet the Press.” “We all make mistakes.”

On Sunday, Mr. Sokolich said he took Mr. Christie at his word when the governor came to Fort Lee on Friday to personally apologize to him. But Mr. Sokolich added that the investigation is still ongoing.

“He was candid. It was, I thought, a productive conversation,” Mr. Sokolich said on “Meet the Press.” “There’s just a lot of stuff there, though.”